To reduce a compound number to the lowest denomination contained in it, multiply the highest by so many as one of this denomination makes of the next lower, and to the product add the number belonging to the next lower; proceed with each succeeding denomination in a similar manner, and the last sum will be the number required.

To reduce a number from a lower denomination to a higher, divide by so many as it takes of this lower denomination to make one of the higher, and the quotient will be the number of the higher; which may be further reduced in the same manner if there are still higher denominations, and the last quotient together with the several remainders will be equivalent to the number to be reduced.

Examples for practice.

In 59lb. 13dwt. 5gr. how many grains?
In 8012131 grains how many pounds, &c.?

Ans. 340157.

Ans. 1390lb. 11oz. 18dwt. 19gr.

In 121. Os. 9 d. how many half pence?
In 58099 half pence how many pounds &c.?

Ans. 58099.

Ans. 121. Os. 9 d.

In 48 guineas at 28s. each how many 4 pence?

Ans. 3584.

In one year of 365d. 5h. 48′ 48′′ how many seconds?

Ans. 31556928.

102. When we have occasion to make use of a number consisting of several denominations as an abstract number, instead of reducing the several parts to the lowest denomination contained in it, we may reduce all the lower denominations to a fraction of the highest. Taking the sum before used, namely, 41. 15s. 9d. we reduce the lower denominations to the higher, as in the last article by division. The number of peace 9, or, is divided by 12, by multiplying the denominator by this number (54), we have thus, s. which being added to 15s. or 1s. the whole number being reduced to the form of a fraction of the same denominator, we have and, which being added, make 113,00 189. This is further reduced to pounds by dividing it by 20, that is, by multiplying the denominator by 20 (54), which

180

1149

gives . Whence £4. 15s. 9d. is equal to £41, or £. This may now be used like any other fraction, and the value of the result found in the different denominations. If we multiply it by 37, we shall have £13, or £177; and £3, reduced to shillings by multiplying the numerator by 20, or dividing the denominator by this number, gives s. or 2s. or 2s. 9d.

240

From the above example we may deduce the following general rules, namely,

To reduce the several parts of a compound number to a fraction of the highest denomination contained in it, make the lowest term the numerator of a fraction, having for its denominator the number which it takes of this denomination to make one of the next higher, and add to this the next term reduced to a fraction of the same denomination, then multiply the denominator of this sum by so many as make one of the next denonination, and so on through all the terms, and the last sum will be the fraction required.†

To find the value of a fraction of a higher denomination in terms of a lower, multiply the numerator of the fraction by so many as make one of the lower denomination, and divide the product by the denominator, and the quotient will be the entire number of this denomination, the fractional part of which may be still further reduced in the same manner.

To reduce 2w. 1d. 6h. to the fraction of a month.

36

6h. is of a day, and being added to one day, or d. gives d. the denominator of which being multiplied by 7, it becomes w. and being added to 2 weeks or twice 13w. gives 389w. If we now multiply the denominator of this by 4, we shall have of a month, as an equivalent expression for 2w. 1d. 6h.

168

† It will often be found more convenient to reduce the several parts of the compound number to the lowest denomination, as by the preceding article for a numerator, and to take for the denominator so many of this denomination as it takes to make one of that, to which the expression is to be reduced; thus 4l. 15s. 9d. being 11494. is equal to 41. because 1d. is l

240

To find the value of of a mile in furlongs, poles, &c.

Reduce 13s. 6d. 2q. to the fraction of a pound.

Ans. £, or £§§.

Reduce 6fur. 26pls. 3yds. 2ft. to the fraction of a mile.

4400

Ans. 14, or 5.

Reduce 7oz. 4pwt. to the fraction of a pound, Troy.

What part of a mile is 6fur. 16pls.?

What part of a hogshead is 9 gallons?

What part of a day is

of a month?

What part of a penny is
What part of a cwt. is

of a pound?
of a pound, Avoirdupois?

What part of a pound is of a farthing?
What is the value of
What is the value of

What is the value of
What is the value of

of a pound, Troy?
of a pound, Avoirdupois?

of a cwt.?
of a mile?

Ans.

3

Ans. z

Ans. 1540·

Ans. 7oz. 4dwt.

Ans. 9oz. 2 dr.

Ans. 3qrs. 3lb. 1oz. 123dr.

Ans. 1fur. 16pls. 2yds. 1ft. 9,2 in.

What is the value of of a day?

900

Ans. 12h. 55' 23"

The several parts of a compound number may also be reduced to the form of a decimal fraction of the highest denomination contained in it, by first finding the value of the expres sion in a vulgar fraction, as in the last article, and then reducing this to a decimal, or more conveniently by changing the terms to be reduced into decimal parts, and dividing the numerator instead of multiplying the denominator by the numbers successively employed in raising them to the required denomination. If we take the sum already used, namely, £4. 15s. 9d. the pence, 9, may be written , or f, the numerator of which admits of being divided by 12 without a remainder. It is thus reduced to shillings and becomes s. or 0,75s. which added to the 15s. makes 15,75s. or reducing the 15 to the same denomination, 1575 or 1575; and this is reduced to pounds, by dividing it by 20, the result of which is 75%, or 0,7875. 4l. 15s. 9d. therefore may be expressed in one denomination, thus, 4,78751. and in 'this state it may be used like any other number consisting of an entire and fractional part. If it be multiplied by 37, we shall have for the product 177,13751. This decimal of a pound may be reduced to shillings and pence, by reversing the above process, or by multiplying successively by 20 and then by 12.*

Тоб

10000

75

0,1375
20

7375

2,7500

12

9,0000

The product therefore of 4l. 15s. 9d. by 37 is 1771. 2s. 9d. as before obtained.

The operation, just explained, admits of a more convenient disposition, as in the following example.

To reduce 19s. 3d. 3q. to the decimal of a pound.

1000

Proceeding as before, we reduce the farthings, 3, considered asq. to hundredths of a penny by dividing by the figure on the left, 4, and place the quotient, 75, as a decimal on the right of the perce; we then take this sum, considered as 3d. or 375gd. that is, annexing as many ciphers as may be necessary, and divide it by 12, which brings it into decimals of a shilling. Lastly, the shillings and parts of a shilling, 19,3125s. considered as 10312500s. are reduced to decimals of a pound by dividing by 20, which gives the result above found.

We may proceed in a similar manner with other denominations of money and with those of the several weights and measOne example in these will suffice as an illustration of the

ures.

method.

To reduce 17pls. 1ft. 6in. to the decimal of a mile.

The decimal in this, as in many other cases, becomes periodical. (97).

From what has been said, the following rules are sufficiently evident. To reduce a number from a lower denomination to the decimul of a higher, we first change it, or suppose it to be changed into a fraction, having 10, or some multiple of 10, for its denominator, and divide the numerator by so many as make one of this higher denomination, and the quotient is the required decimal; which, together with the whole number of this denomination, may again be converted into a fraction, having 10 or a multiple of 10 for its denominator, and thus by division be reduced to a still higher name, and so on.

Also, to reduce a decimal of a higher denomination to a lower, we multiply it by so many as one makes of this lower, and those figures which remain on the left of the comma, when the proper number is separated for decimals (91), will constitute the whole number of this denomination, the decimal part of which may be still further reduced, if there be lower denominations, by multiplying it by the number which one makes of the next denomination, and so on.